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homogeneous system of parameters
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(Definition)
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Let $k$ be a field, let $R$ be an $\mb{N}^m$ -graded $k$ -algebra, and let $M$ be a $\Z^m$ -graded $R$ -module.
Let $\mathcal{H}(R_+)$ be the homogeneous union of the irrelevant ideal of $R$ .
A partial homogeneous system of parameters for $M$ is a finite sequence of elements $\theta_1, \theta_2, \ldots, \theta_r\in\mathcal{H}(R_+)$ such that
where $\dim$ gives the Krull dimension.
A (complete) homogeneous system of parameters is a partial homogeneous system of parameters such that $r=\dim(M)$ .
A sequence $\theta_1,\ldots,\theta_r\in\mathcal{H}(R_+)$ is a homogeneous $M$ -sequence if for all $i$ with $0\leq i<r$ , we have that $\theta_{i+1}$ is not a zero-divisor in
Finally, view $M$ as being $\Z$ -graded by using any specialization of the above $\Z^m$ -grading. Then we define the depth of $M$ to be the length of the longest homogeneous $M$ -sequence.
- 1
- Richard P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhauser Press. Boston, MA. 1986.
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"homogeneous system of parameters" is owned by mathcam.
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| Also defines: |
partial homogeneous system of parameters, complete homogeneous system of parameters, homogeneous  -sequence, depth, depth of a module |
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Cross-references: homogeneous, length, specialization, sequence, Krull dimension, finite sequence, irrelevant ideal, homogeneous union, field
There are 15 references to this entry.
This is version 2 of homogeneous system of parameters, born on 2004-03-12, modified 2004-03-12.
Object id is 5695, canonical name is HomogeneousSystemOfParameters.
Accessed 9764 times total.
Classification:
| AMS MSC: | 13A02 (Commutative rings and algebras :: General commutative ring theory :: Graded rings) |
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Pending Errata and Addenda
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